where \({\bf m}={\bf M}/M^s\)  is the unity magnetization, \({\bf B}_\textrm {eff}\)  is the effective magnetic field, \(\eta _0\) 
is the bulk Gilbert damping, and \({\mathbf {\tau }}=(\gamma /M^s V_m)({\bf m}_\perp \times {\bf I}_{SP}\times {\bf m}_\perp )\) 
is a torque due to the angular momentum
leaking to the NM side. The LLG equation preserves the magnetization modulus; however, since the damping
is small, we will consider an almost constant longitudinal magnetization component, while
\(\dot{{\bf m}}\approx \dot{{\bf m}}_\perp =\dot{\bf m}^y+\dot{\bf m}^z\) . Another approach, which takes into account different damping
from transverse and longitudinal magnetization components, is given by the Lifshitz-Landau-Bloch-Bloembergen (LLBB) equation
[1]}, [2]}, [3]}, [4]}. Provided minor corrections, the SCHA method can also be applied to the LLBB
equation as well. For a thick magnetic film, the injected spin current is related to the magnetization dynamics via
\({\bf I}_{SP}=\frac{\hbar }{4\pi }\left(G^{\uparrow \downarrow }_r{\bf m}\times \dot{\bf m}-G^{\uparrow \downarrow }_i\dot{\bf m}\right),\) 
